| L(s) = 1 | + 5-s − 7-s + 2·11-s + 5·17-s + 2·19-s + 2·23-s − 4·25-s + 10·29-s − 35-s + 5·37-s + 3·41-s − 7·43-s − 3·47-s + 49-s + 6·53-s + 2·55-s + 59-s − 6·61-s + 4·67-s + 8·71-s + 10·73-s − 2·77-s − 3·79-s − 13·83-s + 5·85-s − 6·89-s + 2·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s + 0.603·11-s + 1.21·17-s + 0.458·19-s + 0.417·23-s − 4/5·25-s + 1.85·29-s − 0.169·35-s + 0.821·37-s + 0.468·41-s − 1.06·43-s − 0.437·47-s + 1/7·49-s + 0.824·53-s + 0.269·55-s + 0.130·59-s − 0.768·61-s + 0.488·67-s + 0.949·71-s + 1.17·73-s − 0.227·77-s − 0.337·79-s − 1.42·83-s + 0.542·85-s − 0.635·89-s + 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.706383893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.706383893\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 + 13 T + p T^{2} \) | 1.83.n |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.03730592756073252170993163438, −9.731110109277065823801604813257, −8.679762585429044174907509692288, −7.77797671333296440572234908419, −6.75309219456590577358633993250, −5.97440167341892660090228386416, −5.01523418584733424644218383591, −3.78800010230879522821556973826, −2.72932154679755710467964549500, −1.19245405811260080299254930543,
1.19245405811260080299254930543, 2.72932154679755710467964549500, 3.78800010230879522821556973826, 5.01523418584733424644218383591, 5.97440167341892660090228386416, 6.75309219456590577358633993250, 7.77797671333296440572234908419, 8.679762585429044174907509692288, 9.731110109277065823801604813257, 10.03730592756073252170993163438
